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Table 2 Summary statistics of the performance under scenario 1 with the true parameter of DRF = 2

From: A doubly robust estimator for continuous treatments in high dimensions

  

N = 500, P = 200

N = 200, P = 100

  

DRF

Est_

Est_

Est_

Boot_

Boot_

Boot_

emp_

RMSE

DRF

Est_

Est_

Est_

Boot_

Boot_

Boot_

emp_

RMSE

Std

Coverage

Power

std

coverage

power

std

Std

Coverage

Power

std

coverage

power

std

SoSt

rho = 0

GOALDeR

2.009

0.025

0.85

1

0.041

0.98

1

0.033

0.034

2.01

0.034

0.9

1

0.061

1

1

0.038

0.04

SL-DR

2.002

0.037

0.79

1

-

-

-

0.087

0.087

2.021

0.039

0.63

1

-

-

-

0.128

0.129

GOAL

2.012

0.1

1

1

-

-

-

0.052

0.053

2.025

0.162

1

1

-

-

-

0.106

0.109

SoWt

rho = 0

GOALDeR

1.995

0.027

0.9

1

0.048

1

1

0.031

0.031

1.986

0.04

0.87

1

0.065

0.98

1

0.047

0.048

SL-DR

1.997

0.047

0.8

1

-

-

-

0.125

0.124

1.994

0.049

0.79

1

-

-

-

0.102

0.102

GOAL

1.998

0.087

0.99

1

-

-

-

0.056

0.056

2.004

0.151

1

1

-

-

-

0.096

0.096

WoSt

rho = 0

GOALDeR

2.012

0.022

0.87

1

0.041

1

1

0.028

0.031

1.998

0.033

0.81

1

0.061

0.98

1

0.051

0.051

SL-DR

2.02

0.033

0.78

1

-

-

-

0.073

0.075

1.995

0.032

0.64

1

-

-

-

0.06

0.06

GOAL

2.002

0.087

0.96

1

-

-

-

0.064

0.063

2.002

0.133

0.97

1

-

-

-

0.095

0.095

SoSt

rho = 0.2

GOALDeR

2.002

0.021

0.83

1

0.04

0.98

1

0.032

0.032

2.001

0.031

0.75

1

0.061

0.97

1

0.056

0.055

SL-DR

2.004

0.033

0.57

1

-

-

-

0.063

0.063

2.017

0.043

0.66

1

-

-

-

0.119

0.12

GOAL

2.003

0.148

0.99

1

-

-

-

0.117

0.116

2.066

0.185

0.97

1

-

-

-

0.132

0.147

SoWt

rho = 0.2

GOALDeR

1.985

0.025

0.9

1

0.046

1

1

0.027

0.031

1.982

0.038

0.78

1

0.066

0.98

1

0.052

0.055

SL-DR

1.999

0.04

0.81

1

-

-

-

0.061

0.061

2.004

0.047

0.7

1

-

-

-

0.098

0.097

GOAL

2.01

0.13

0.99

1

-

-

-

0.072

0.073

1.998

0.2

1

1

-

-

-

0.117

0.116

WoSt

rho = 0.2

GOALDeR

2.003

0.021

0.9

1

0.038

1

1

0.025

0.025

1.989

0.031

0.81

1

0.062

0.98

1

0.044

0.045

SL-DR

1.992

0.033

0.8

1

-

-

-

0.055

0.055

1.968

0.04

0.61

1

-

-

-

0.123

0.126

GOAL

2.01

0.122

0.99

1

-

-

-

0.098

0.098

2.068

0.159

0.95

1

-

-

-

0.116

0.134

SoSt

rho = 0.5

GOALDeR

1.983

0.024

0.81

1

0.041

0.96

1

0.034

0.038

1.984

0.031

0.67

1

0.066

0.98

1

0.059

0.061

SL-DR

1.986

0.023

0.62

1

-

-

-

0.039

0.041

1.98

0.027

0.52

1

-

-

-

0.062

0.065

GOAL

2.046

0.183

0.99

1

-

-

-

0.146

0.152

2.112

0.223

0.9

0.99

-

-

-

0.213

0.24

SoWt

rho = 0.5

GOALDeR

1.973

0.029

0.85

1

0.047

0.98

1

0.029

0.04

1.955

0.039

0.68

1

0.071

0.95

1

0.056

0.072

SL-DR

1.983

0.034

0.64

1

-

-

-

0.08

0.082

1.963

0.037

0.59

1

-

-

-

0.07

0.079

GOAL

2.005

0.186

0.99

1

-

-

-

0.114

0.114

2.005

0.186

0.99

1

-

-

-

0.114

0.114

WoSt

rho = 0.5

GOALDeR

1.979

0.024

0.74

1

0.04

0.96

1

0.034

0.04

1.977

0.032

0.74

1

0.065

0.97

1

0.056

0.06

SL-DR

1.987

0.025

0.77

1

-

-

-

0.039

0.041

1.971

0.026

0.48

1

-

-

-

0.065

0.07

GOAL

2.045

0.16

0.98

1

-

-

-

0.136

0.143

2.045

0.16

0.98

1

-

-

-

0.136

0.143

  1. Notations: DRF = 2, the parameter of DRF is 2; rho = 0, rho = 0.2, and rho = 0.5, there are no, moderate and strong correlations between covariates; emp_std: the empirical standard deviation of the estimates; Est_Std: the mean of standard deviation estimates using sandwich-type estimator (GOAL) or regression of the treatment on pseudo-outcome (GOALDeR and SL-DR); Boot_Std: the mean of bootstrapped standard deviation; coverage: coverage probability of the 95% confidence interval; RMSE: root mean square error calculated as \(\:\text{R}\text{M}\text{S}\text{E}=\sqrt{\frac{1}{B}{\sum\:}_{b=1}^{B}{\left({\widehat{\eta\:}}_{b}-\eta\:\right)}^{2}}\)
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BMC Medical Research Methodology

ISSN: 1471-2288

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